How to enable Vertex Color Paint in v2.81.16? Can't find material options in new UI
First the video, then some explanation:


In the top left corner of your 3D View window, there you can find a Combo Box with the choice for "Object Mode" usually the one you start in, "Edit Mode" and further down "Vertex Paint Mode". That said, materials, you find on the right side down, in the "Properties Panel", all the way down. If you ask for material related settings in the render settings, 2nd from top, then you will have to consider which render-engine you want to use. Evee and Cycles have "some" specific options, while most materials you can set up with the nodes can be used with both. In case something does not work, you can build one material with 2 output nodes, one for Evee and one for cycles.